What question did this study set out to answer?

To investigate the arithmetic-geometric dichotomy in the Erdős-Szemerédi sum-product conjecture and formally verify foundational theorems in additive combinatorics.

August 20, 2026Open Access

On the Erdős-Szemerédi Sum-Product Conjecture

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CNCharles EDOU NZE

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Implication

Theoretical analysis establishes lower bounds and structural proofs for sumsets and product sets, highlighting machine-checked additive combinatorics in Lean 4.

Key Points

To investigate the arithmetic-geometric dichotomy in the Erdős-Szemerédi sum-product conjecture and formally verify foundational theorems in additive combinatorics.
Analyzed discrete transversal chains, Freiman-type difference sets, and Cauchy-Schwarz additive energy bounds.
Mapped sum-product bounds to point-line incidence geometry in R^2 using the Szemerédi-Trotter theorem to derive Elekes' bound.
Implemented complete, machine-checked formal proofs of mathematical derivations using the Lean 4 interactive theorem prover.
Proved the exact discrete lower bound |A + A| ≥ 2n - 1 for finite sets of size n, demonstrating equality holds if and only if A forms an exact arithmetic progression.
Derived the reciprocal additive energy bound |A + A| ≥ |A|^4 / E_+(A) connecting sumset expansion directly to representation functions.
Formally verified Elekes' geometric bound max(|A+A|, |A · A|) ≥ c|A|^(5/4) with 100% machine-checked proofs hosted in a public repository.

Cite This Study

Charles EDOU NZE (2026) studied this question.

synapsesocial.com/papers/6a86b5eb8a91293e6a1cd8a9 https://doi.org/10.5281/zenodo.21992212

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Implication

Theoretical analysis establishes lower bounds and structural proofs for sumsets and product sets, highlighting machine-checked additive combinatorics in Lean 4.

On the Erdős-Szemerédi Sum-Product Conjecture

Discussion

Member takes

Implication

Key Points

Cite This Study

On the Erdős-Szemerédi Sum-Product Conjecture

Discussion

Member takes

Implication

Key Points

Cite This Study

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